Barriers in impulsive antiperiodic problems

Barriers in impulsive antiperiodic problems

Irena Rach ˚unková

and Lukáš Rach ˚unek

1Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czechia

2Department of Algebra and Geometry, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czechia

Received 3 June 2019, appeared 10 November 2019 Communicated by Josef Diblík

Abstract. Some real world models are described by means of impulse control of non- linear BVPs, where time instants of impulse actions depend on intersection points of solutions with given barriers. For i = 1, . . . ,m, and [a,b] ⊂ R, continuous functions γ_i :R →[a,b]determine barriers Γi = {(t,z) :t =γ_i(z),z∈ R}. A solution(x,y)of a planar BVP on [a,b] is searched such that the graph of its first component x(t) has exactly one intersection point with each barrier, i.e. for eachi∈ {1, . . . ,m}there exists a unique roott= t_ix ∈ [a,b]of the equationt=γ_i(x(t)). The second componenty(t) of the solution has impulses (jumps) at the pointst_1x, . . . ,tmx. Since a size of jumps and especially the pointst1x, . . . ,tmx depend onx, impulses are calledstate-dependent.

Here we focus our attention on an antiperiodic solution (x,y) of the van der Pol equation with a positive parameterµand a Lebesgue integrable antiperiodic function f

x⁰(t) =y(t), y⁰(t) =µ

x(t)−^x

3(t) 3

⁰

−x(t) +f(t) for a.e.t∈R, t6∈ {t1x, . . . . ,tmx},

whereyhas impulses at the points from the set{t_1x, . . . ,tmx}, y(t+)−y(t−) =J_i(x), t=tix, i=1, . . . ,m, andJ_iare continuous functionals defining a size of jumps.

Previous results in the literature for this antiperiodic problem assume that impulse points are values of given continuous functionals. Such formulation is certain handicap for applications to real world problems where impulse instants depend on barriers. The paper presents conditions which enable to find such functionals from given barriers.

Consequently the existence results for impulsive antiperiodic problem to the van der Pol equation formulated in terms of barriers are reached.

Keywords: van der Pol equation, state-dependent impulses, barriers, existence, an- tiperiodic solution.

2010 Mathematics Subject Classification: 34A37, 34B37.

BCorresponding author. Email: irena.rachunkova@upol.cz

1 Introduction

Some models of real world problems are characterized by the occurrence of abrupt changes of their behavior at certain time instants depending on the state and situation of a model. A nat- ural assumption for differential models is that these instants (impulse points) are determined by means of intersections of a solution of a model with given barriers. For periodic problems see [1,2,8,9,14,16] and for boundary value problems with various linear boundary conditions see [10] and [15]. Existence theorems in these papers are not applicable to equations of van der Pol type. On the other hand we can find existence theorems for impulsive periodic or antiperiodic solutions to equations of van der Pol type, but these results are proved under the assumption that impulse points are values of given continuous functionals [3–6,11–13]. This brings difficulties in applications where impulse instants depend on barriers.

The aim of this paper is to overcome this handicap. In particular:

• For positive numbersKandL, an appropriate function setΩKL (see (2.1)) is determined.

• Conditions for barriersΓi ={(t,z)_:t= γ_i(z)_,z∈_R}_,i=_{1, . . . ,}m, are found such that a graph of each functionx∈ _ΩKL has exactly one intersection point (t_ix,x(t_ix))with each of the barriers (see Lemma2.2).

• The conditions imply in addition that pointst_ix depend continuously onx (see Lemma 2.3).

• Conditions formulated in terms of barriers and guaranteeing the solvability of an im- pulsive antiperiodic problem to the van der Pol equation are found (see Theorem1.1).

More precisely, for T > 0 and given continuous functions γ₁, . . . ,γ_m, we prove the exis- tence of aT-antiperiodic solution(x,y)of the van der Pol equation with a positive parameter µand a Lebesgue integrable T-antiperiodic function f

x⁰(t) =y(t), y⁰(t) =µ

x(t)− ^x

3(t) 3

0

−x(t) + f(t) ^{for a.e.t}∈[0,T], t 6∈ {t_1x, . . . . ,tmx}, (1.1) wherey has impulses at the pointst_1x, . . . ,t_mx ∈ (_0,T)determined by the barriersΓ1, . . . ,Γm

through the equalities

t_ix =γ_i(x(t_ix)), i=1, . . . ,m, (1.2) andyis continuous anywhere else in [0,T]. The impulse conditions have the form

y(t+)−y(t−) =J_i(x), t =t_ix, i=1, . . . ,m, (1.3) whereJ_i are continuous bounded functionals defining a size of jumps.

Notations

T-antiperiodic function x (satisfying (1.1), (1.2), (1.3)) will be found in the set of 2T-periodic real-valued functions. To do it functional sets defined below are used.

• L¹ consists of 2T-periodic Lebesgue integrable functions on [0, 2T] with the norm kxk_L1 := _2T¹ R2T

0 |x(t)|dt,

• BV consists of 2T-periodic functions of bounded variation on[0, 2T],

• var(x)for x∈BV is the total variation ofxon [0, 2T],

• kxk_{∞ :}=_sup{|x(t)|_:t ∈[0, 2T]}_forx∈_BV,

• NBV consists of normalized functionsx∈ BV in the sense thatx(t) = ¹₂(x(t+) +x(t−)),

• ¯x:= _2T¹ R2T

0 x(t)dt=0 is the mean value of x∈BV,

• NBV consists from functions] x ∈ NBV with ¯x = 0; NBV with the norm var] (x) is the Banach space,

• AC(J) consists of 2T-periodic absolutely continuous functions on J ⊂ [0, 2T] and if J = [0, 2T]we write AC,

• AC :g =AC∩NBV.]

• A couple (x,y) ∈ ACg×NBV satisfying (1.1), (1.2), (1.3) is a 2T-periodic] solution of problem (1.1)–(1.3). If in addition

x(0) =−x(T), y(0) =−y(T), (1.4) then(x,y)is aT-antiperiodic solutionof problem (1.1)–(1.3).

Figure 1.1: The first component xof T-antiperiodic solution(x,y)of a problem with two barriers Γ1andΓ2

The main existence result is contained in the next theorem.

Theorem 1.1 (Main result). Let T ∈ (0,√

3), K,L ∈ (0,∞), let J_i, i = 1, . . . ,m, be contiuous bounded functionals onNBV, and let f] ∈L¹ be T-antiperiodic, i.e. f(t+T) =−f(t)for a.e. t ∈_R.

Assume that there exist a,b∈ (0,T)such that functionsγ₁, . . . ,γ_m satisfy

0< a≤γ₁(z)<γ₂(z)<· · ·<γ_m(z)≤b<T, z ∈[−K,K]_{. (1.5)}

Further, assume that L_i ∈(0, 1/L), i =1, . . . ,m, are such that

|γ_i(z₁)−γ_i(z₂)| ≤ L_i|z₁−z₂|, z₁,z₂ ∈[−K,K], i=1, . . . ,m. (1.6) Then there exists µ₀ > 0such that for each µ ∈ (0,µ₀]problem(1.1)–(1.3) has a T-antiperiodic solution(x,y), where y has m jumps at the points t_1x, . . . ,t_mx ∈ [a,b]and y is continuous anywhere else in[0,T]. Moreover the estimate

|x(t)| ≤var(x)≤K, |y(t)| ≤L, t∈[0,T], (1.7) is valid.

We can find the optimal (maximal) valueµ₀as follows. SinceJ_i are bounded, it holds J_i :NBV] →[−a_i,a_i], i=1, . . . ,m,

for somea_i ∈(0,∞). Denote

c₁ := Tkfk_L1+

∑

a_i, (1.8)

and define a functionϕby

ϕ(µ):= ¹−µT− ^T₃² 3

s

1−µT− ^T₃²

µT , µ∈(0, 1/T−T/3]. (1.9) Then, according to the proof of Theorem1.1, µ₀ = ϕ⁻¹(Tc₁)∈(0, 1/T−T/3).

Auxiliary results Denote

(x∗y)(t):= ¹ 2T

Z _2T

0 x(t−s)y(s)ds, t ∈[0, 2T] forx,y ∈L¹, and remind the inequalities

var(x∗y)≤var(x)kyk_∞, x,y∈NBV, (1.10) var(x∗f)≤_var(x)kfk_L1, x ∈_NBV, f ∈_L¹_{, (1.11)} kxk_L1 ≤ kxk_∞ ≤_var(x)_, x∈_NBV.] _(1.12) Further, using the function

E₁(t) =

(T−t fort∈(0, 2T), 0 fort=0, which fulfils

var(E₁) =4T, kE₁k_∞ =T, (1.13)

we introduce antiderivative operatorsI andI² by

Iu:= E₁∗u∈gAC, I²u:= I(Iu)∈ gAC, u∈L¹. (1.14) Forτ∈_Rwe define a distributionε_τ by the Fourier series

ε_τ :=

∑

n∈_Z

(1−(−1)ⁿ)e^{inπT (t−τ)}, t ∈_R. (1.15)

Then it holds

Iετ ∈_NBV,] _I²_ετ ∈gAC, kIετk_∞= T. (1.16) See [11] for more details. Using this we investigated in [11] the van del Pol equation

x⁰(t) =y(t), y⁰(t) =µ

x(t)−^x

3(t) 3

0

−x(t) + f(t) for a.e.t ∈_R, (1.17) with a positive parameter µ, a Lebesgue integrable T-antiperiodic function f, and with the state-dependent impulse conditions

t→limτ_i(x)+y(t)− lim

t→τ_i(x)−y(t) =J_i(x), i=1, . . . ,m, (1.18) where J_i and alsoτ_i,i=1, . . . ,m, are given continuous and bounded real-valued functionals on NBV. For such setting we proved the existence result contained in Theorem] 1.2.

Theorem 1.2 ([11, Theorem 1.1]). Assume that T ∈ (_0,√

3), and the functionals τ₁, . . . ,τ_m have values in(0,T). Further, let

i6= j =⇒ τ_i(x)6=τ_j(x), x∈AC,g i,j=1, . . . ,m. (1.19) Then there existsµ0> 0such that for eachµ∈(0,µ0]the problem (1.17),(1.18)has a T-antiperiodic solution(x,y).

2 Existence of continuous functionals

If we study an impulsive boundary value problem which is formulated by means of barriers Γ1, . . . ,Γm, then a number of impulse points for some solution(x,y)is equal to a number of values of t satisfying the equations t−γ_i(x(t)) =0, i = 1, . . . ,m. In general, such equations need not be solvable, or they can have finite or infinite number of roots. In Theorem 1.1 we present conditions imposed on barriers which yield for each i∈ {1, . . . ,m}a unique solution t=t_ixof the equation t=γ_i(x(t))provided xbelongs to some suitable set ΩKL.

For positive numbersKandL, we define a set ΩKL

ΩKL :={x∈ gAC : var(x)≤K, |x⁰(t)| ≤ Lfor a.e. t∈[0, 2T], xis T-antiperiodic}, (2.1) and prove its properties.

Lemma 2.1. The setΩKLis nonempty, bounded, convex and closed in_NBV.]

Proof. ΩKL is nonempty because the zero function belongs toΩKL and ifK≤ LT, then x(t) =

K

4 sin(πt/T) ∈ _ΩKL, if K > LT, then x(t) = ^LT₄ sin(πt/T) ∈ _ΩKL. In addition, we see that ΩKL is bounded and convex. It remains to prove that ΩKL is closed. Consider a sequence {x_n}^∞_n=1⊂_ΩKL and let x∈NBV is such that]

nlim→_∞var(x−x_n) =0. (2.2)

We need to prove thatx∈ _ΩKL. From var(xn)≤K,n∈N, and (2.2) it follows that var(x)≤K.

Further, there exists a unique function x^AC ∈ gAC such that x = x^AC+x^S, where x^S ∈ _NBV]

is a singular part of x having zero derivative for a.e. t ∈ [0, 2T]. Moreover, since x_n ∈ AC,g n∈N, we have by [7, Theorem 3.3.5],

var(x−x_n) =var(x^AC−x_n) +var(x^S), n∈_N,

and lettingn → _{∞, we get} x^S ≡ 0 due to (2.2). Consequently x ∈ AC and there exists a setg M ⊂ (0, 2T) of a zero measure such that x⁰(t) is defined for all t ∈ (0, 2T)\M. Choose an arbitraryt ∈ (0, 2T)\M. We can findε>0 such that (t−_ε,t+ε)⊂(0, 2T). Having in mind that|x⁰_n(t)| ≤Lfor a.e. t∈ [0, 2T]and alln∈N, we get forh∈ (−ε,ε)

|xn(t+h)−xn(t)| ≤

Z _t+h

t

|x_n⁰(s)|ds

≤ L|h|.

This yields |x(t+h)−x(t)| ≤ L|h|, and after the limit h → 0 we get |x⁰(t)| ≤ L for a.e.

t ∈ [0, 2T]. Finally, for each n ∈ N, the function x_n isT-antiperiodic which implies by (1.12)

and (2.2) thatxisT-antiperiodic, as well.

Lemma 2.2. Let K,L∈(0,∞). Assume that there exist a,b∈ (0,T)and L_i ∈(0, 1/L), i=1, . . . ,m, such that(1.5)and(1.6)are fulfilled. Then for each x∈ _ΩKL and i∈ {1, . . . ,m}the equation

t= γ_i(x(t)) (2.3)

has a unique solution t_ix ∈[a,b].

Proof. Choose x ∈ _ΩKL, i ∈ {1, . . . ,m}, and put σ_x(t) = t−γ_i(x(t)) for t ∈ [0,T]. Then

|x|_∞ ≤K,σx is continuous and by (1.5),σx(0)<0,σx(T)>0. This yieldstx∈ (0,T)such that σ_x(t_x) =0. Let t_x,s_x ∈ (0,T)satisfyγ_i(x(t_x)) =t_x,γ_i(x(s_x)) =s_x. Then, by (1.6) and (2.1),

|sx−tx|=|γ_i(x(sx))−γ_i(x(tx))| ≤L_i|x(sx)−x(tx)|

=L_i

Z _sx

tx

x⁰(ξ)dξ

≤L_iL|sx−tx|<|sx−tx|,

which givest_x =s_x.

Lemma 2.3. Let the assumptions of Lemma2.2be fulfilled. Then for i ∈ {1, . . . ,m}, the functional τ_i :ΩKL →[a,b], τ_i(x) =t_ix, (2.4) where t_ixis a solution of (2.3), is continuous.

Proof. Choosex,v∈_ΩKL andi∈ {1, . . . ,m}. Then

|τ_i(x)−τ_i(v)|= |t_ix−t_iv|=|γ_i(x(t_ix))−γ_i(v(t_iv))| ≤ L_i

|x(t_ix)−v(t_ix)|+|

Z _tix

t_iv v⁰(ξ)dξ|

, and so

|τ_i(x)−τ_i(v)| ≤L_ivar(x−v) +L_iL|t_x−t_v|) =L_ivar(x−v) +L_iL|τ_i(x)−τ_i(v)|. Therefore

|τ_i(x)−τ_i(v)| ≤ ^Livar(x−v)

1−L_iL , x,v∈_ΩKL,

which yields the continuity ofτ_i onΩKL.

3 Proof of Theorem 1.1

Proof. (i) Having continuous functionalsτ₁, . . . ,τ_m from Lemma2.3, we can argue similarly as in [11] because (1.5) implies (1.19). SinceJ_i are bounded, there exista_i ∈ (_0,∞)_{, i}=_{1, . . . ,m,} such that

J_i :NBV] →[−a_i,a_i], i=1, . . . ,m. (3.1) Choose

T∈ (_0,√

3)_, µ∈(0, 1/T−T/3)_{, (3.2)}

introduce constantsK,Lby K:= ¹

2 s

1−µT−T²/3

µT , (3.3)

L:= µK+ ^2µ

3 K³+TK+2Tkfk_L1 +¹ 2

∑

a_i, (3.4)

and consider the setΩKL from (2.1). Similarly as in [11] we define an operator F _by Fx=µI

x− ^x3

3

+I² −x+ f+ ¹ 2T

∑

J_i(x)ε_τi(x)

!

, x∈_{ΩKL, (3.5)} whereτ_iis from (2.4),ε_τi(x)is from (1.15),I,I²are from (1.14). It follows from [11, Lemma 4.2]

that F is compact onΩKL.

(ii) Let us show that F maps ΩKL to ΩKL. Since the definition of the set ΩKL in (2.1) is different from the definition of the corresponding setΩin [11], we need to prove the estimate

|(Fx)⁰(t)| ≤L for a.e.t∈[0, 2T], and allx ∈_ΩKL. (3.6) Differentiating (3.5) we get

(Fx)⁰(t) =µ x(t)− ^x

3(t)

3 −x¯+ ^x

3

3

!

+I(f(t)−x(t)) + ¹ 2T

∑

J_i(x)Iε_τi(x) (t), and, by

|(Fx)⁰(t)| ≤µ|x(t)|+ ^µ

3|x³(t)−x³|+var(E₁∗(f−x)) + ¹ 2T

∑

|J_i(x)|kIε_τi(x)k_∞

≤µkxk_∞+ ^2µ

3 kxk³_∞+var(E₁)kfk_L1+kE₁k_∞var(−x) + ¹ 2T

∑

a_ikIε_τi(x)k_∞

≤µK+^2µ

3 K³+2Tkfk_L1+TK1 2

∑

a_i = L.

Now, considerc₁and ϕfrom (1.8) and (1.9) and assume that

Tc₁ ≤ ϕ(µ). (3.7)

Then, using the arguments from the proof in [11, Theorem 4.4], we get

var(Fx)≤ K for all x∈_{ΩKL. (3.8)}

In addition, by (1.16) and (1.14), Fx ∈ AC and it is antiperiodic forg x ∈ _ΩKL. Therefore F(_ΩKL)⊂_ΩKL.

(iii) Consequently, by the Schauder fixed point theorem there exists a fixed point x∈ _ΩKL of the operatorF. By [11, Lemma 4.1, Lemma 3.4], if we puty(t) = x⁰(t)for a.e. t∈ _{R, then} (x,y)is aT-antiperiodic solution of problem (1.1)–(1.3). Having in mind that ϕis continuous and decreasing on(0, 1/T−T/3]and limµ→0+ϕ(µ) =_∞,ϕ(1/T−T/3) =0, we get a unique µ₀∈(0, 1/T−T/3)satisfying Tc₁ = ϕ(µ₀). Clearly, ifµ≤ µ₀, then (3.7) holds. Consequently we get aT-antiperiodic solution of problem (1.1)–(1.3) for eachµ∈ (0,µ₀].

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